Realization of quasicrystalline quadrupole topological insulators in electrical circuits

Quadrupole topological insulators are a new class of topological insulators with quantized quadrupole moments, which support protected gapless corner states. The experimental demonstrations of quadrupole-topological insulators were reported in a series of artificial materials, such as photonic crystals, acoustic crystals, and electrical circuits. In all these cases, the underlying structures have discrete translational symmetry and thus are periodic. Here we experimentally realize two-dimensional aperiodic-quasicrystalline quadrupole-topological insulators by constructing them in electrical circuits, and observe the spectrally and spatially localized corner modes. In measurement, the modes appear as topological boundary resonances in the corner impedance spectra. Additionally, we demonstrate the robustness of corner modes on the circuit. Our circuit design may be extended to study topological phases in higher-dimensional aperiodic structures. Higher order topological systems add an additional layer of complexity to the bulk boundary correspondence by being able to sustain additional modes such as corner and hinge states. Here, the authors use electrical circuits to realize a two-dimensional quasicrystalline quadrupole topological insulator without discrete translational symmetry and observe localized corner modes.

Highly customizable electrical circuits have emerged as a new platform to engineer various topological states [44][45][46][47][48][49][50][51][52][53][54][55][56][57][58][59] . In this study, we report the realization of the 2D quasicrystalline QTIs in electrical circuits and observe the corner states experimentally. To realize the QTIs, we construct the Ammann-Beenker (AB) tiling circuits with two types (thin and fat) of rhombuses and we implement the nearest-and next-nearest-neighbor hoppings using lumped elements to realize rotational symmetry. Using impedance measurement, the corner states protected by quantized quadrupole moment are directly observed in the circuits. The robustness of topological corner states is also demonstrated. Our work provides the experimental evidence of topological phases in 2D quasicrystalline systems. We expect more topological states can be observed in circuits based on similar implementations.

Results
Quasicrystalline QTIs. We construct the QTI on an AB tiling quasicrystal by only considering nearest-and next-nearestneighbor hoppings 35 . The tight-binding Hamiltonian is where T ϕ jk ¼ and ϕ jk indicates the polar angle of bond between site j and k with respect to the horizontal direction. Here, c y j ¼ c y j1 ; c y j2 ; c y j3 ; c y j4 is the creation operator in cell j. The first and second terms are the intracell and intercell hoppings with amplitudes A 0 and A 1 , respectively. Γ 4 ¼ τ 1 τ 0 and Γ υ ¼ Àτ 2 τ ν with ν = 1,2,3. τ 1,2,3 are the Pauli matrices and τ 0 is the identity matrix. This model has a fourfold rotational symmetry C 4 = UR, where U ¼ 0 iτ 2 τ 0 0 and R is an orthogonal matrix permuting the sites of the tiling to rotate the whole system by π/4. The rotational symmetry C 4 results in a quantized quadrupole moment Q xy = 0,e/2. Thus, Q xy is a natural topological invariant, which can be calculated in real space 60,61 . By numerical calculations, we confirmed that the AB tiling quasicrystal has the non-trivial quadrupole moment Q xy = e/2 under the hopping ratio λ > 2.5(λ = A 1 /A 0 ) (see Supplementary Note 1). The quantized quadrupole moment indicates the occurrence of the quadrupole insulator with protected corner states.
Realization in electrical circuits. The quasicrystalline lattice can be mapped to an electrical circuit. To realize quasicrystalline QTI, we design an electrical circuit depicted in Fig. 1a. The circuit is characterized by the Kirchhoff's law where with C pa,qb being the capacitance between two sites and W pa;qb ¼ L À1 pa;qb being the inverse inductivity between two sites. Here, the subscript g means the ground. H pa,qb can be equivalent to the Hamiltonian of quasicrystal in Eq. (1) if the diagonal elements are zero at ω 0 . For the diagonal components with pa = qb, the grounded elements are chosen for satisfying C pa;pa ¼ ÀC pa;g À Fig. 1a, the solid and dashed lines correspond to hoppings with positive and negative values, which can be realized by the capacitors and inductors, respectively. We choose capacitor C 1 and inductor L 1 for intracell hopping, and C 2 and L 2 for intercell hopping. The nearest-neighbor couplings between cells are introduced to stabilize the corner states in small size systems 35 . If these couplings are negleted, the exact AB tiling is restored and the physical results do not change qualitatively.
Under a suitable choice of the grounded elements, the tightbinding Hamiltonian of the QTI on a quasicrystalline lattice in Eq. (1) is mapped to the circuit Hamiltonian in Eq. (2). It is noteworthy that there exists a relation C 2 /C 1 = L 1 /L 2 = A 1 /A 0 .
The two-point impedance is: where V a(b) is the voltage on site a(b), I ab is the current between the two sites, and j n and ψ n are the eigenvalue and eigenstates of the matrix J(ω), respectively 22 . The two-point impedanace Z ab diverges in the presence of zero-admittance modes (j n = 0) with ψ n;a ≠ψ n;b . Hence, the corner states with zero-admittance can be observed by measuring the two-point impedance.
We consider the QTI on an AB tiling 2D quasicrystalline lattice. A circuit that realizes the QTIs is depicted in Fig. 1a. The unit cell of the circuit contains four sites denoted by labels 1, 2, 3, and 4. Each unit cell consists of three capacitors for positive coupling and one inductor for negative coupling, which can generate a synthetic magnetic π flux threading the unit-cell plaquette (equivalent to half the magnetic flux quantum, where h is the Planck constant). The existence of this non-zero flux opens the spectral gap for maintaining the corner-localized mid-gap modes. We use two pairs of capacitors and inductors (C 1 , L 1 ) and (C 2 , L 2 ), which have the same resonant frequency ω 0 ¼1= ffiffiffiffiffiffiffiffiffi ffi , as the intracell and intercell wirings between the sites, respectively. The intracell and intercell elements are related by C 2 = λC 1 and L 2 = L 1 /λ. The circuit is governed by the linear circuit theory with a circuit Laplacian J(ω) 22 . The circuit has a square-open boundary, which satisfies the global C 4 symmetry at the resonant frequency ω 0 . Suitable grounded elements on the sites are chosen (site colors depicted in Fig. 1a indicate the values of the grounded capacitors and/or inductors) to sustain chiral symmetry at ω 0 , which pins the topological boundary modes in the middle of the bulk energy gap. The symmetry characteristic and quantized quadrupole moment of circuit indicate the occurrence of the QTI associated with topologically protected states localized on the corner sites. The edge states are gapped and merged with the bulk states. Hence, it is difficult to observe them experimentally (see Supplementary Note 2).
To realize the QTI experimentally, a circuit with 69 unit cells was fabricated as shown in Fig. 1b. The intracell elements with C 1 = 10 nF and L 1 = 1 mH result in a resonant frequency 50.3 kHz (parameters for other elements are given in "Methods").
Here we set the coupling ratio λ = 10, to get highly localized corner states, which can be observed by two-point impedance measurements between the corner/edge/bulk site and another bulk site using an impedance analyzer. Figure 2 compares the experimental and theoretical results, which demonstrates the spectral and spatial localizations of the topological corner states. Figure 2a shows the spectrum of the circuit Laplacian J(ω) as a function of the normalized frequency ω/ ω 0 . The isolated corner modes reside in the spectral gap of J(ω) at a fixed frequency ω 0 . The spatial distribution of the experimentally observed impedance of the corner states at ω 0 is illustrated in Fig. 2b. The impedance is maximum at the four corner sites and exponentially decays at other sites. The comparison between the experimental and theoretical impedance spectra is shown in Fig. 2c, d, which demonstrate the spectral localization of the corner modes. The maximum measured impedance reaches 7 kΩ.
To justify the robustness of the corner states, we consider the effect of experimental element errors. The errors are generated in two ways: (i) random manufacturing variations in discrete elements and (ii) the deviation of the typical values of commercially available circuit components from the theoretical circuit parameters. Figure 3a shows the normalized eigenfrequencies ω/ω 0 of the Laplacian J(ω) with element error ±5% and the number of the samples is 300. The eigenfrequencies of corner states (indicated by red circles) are located in the bandgap and far away from those of other states (indicated by blue circles). Experimentally, we construct a circuit with element error ±5% and measure the two-point impedances as a function of the normalized frequency ω/ω 0 . As shown in Fig. 3b, the corner states are spectrally localized with a deviation from ω 0 . Figure 3c-f shows the spatial impedance distributions at the frequencies of the four corner states. Compared with the spatial impedance distribution in Fig. 2b, although the maximum impedances are located at only one of the four corner sites, the topological corner states are still spatially localized. The spectral and spatial localizations imply the robustness of corner states on the circuit.

Discussion
The quadruple insulators were always realized on periodic systems with translational symmetry and the quadruple moments are well-defined in momentum space. Unlike the periodic structures, the quasi-periodic systems possess long-range order but do not have translational symmetry. Therefore, quasicrystalline insulators with a quantized quadrupole moment extend the concept of quadruple insulators defined in crystals. The realization of quadruple isolator in this work opens a way for the implementation of corner states in more quasi-periodic systems.
The circuit implementation of the 2D quasicrystalline QTIs confirms the existence and robustness of the corner states on the circuit. This work provides an experimental evidence for the first time to implement the topological phase in a 2D quasicrystalline system and extends the territory of topological phases beyond crystals to higher-dimensional aperiodic systems. The highly customizable circuit platform can also readily be applied to other 2D quasicrystals with different symmetries and to threedimensional quasicrystalline structures 62 , which may realize more exotic topological states.

Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.